* Sharp rise in militant attacks on state targets in south (Updates with Yemen saying 37 al Qaeda militants killed)

ADEN, Yemen, May 4 (Reuters) - Yemen said 37 al Qaeda militants were killed in the southern province of Shabwa on Sunday as the army intensified an offensive to root out foreign and local Islamist fighters holed up in some of the country's most impenetrable areas.

The report by state news agency Saba came shortly after a military source told Reuters a suicide bomber killed six soldiers by their outpost in Shabwa, one of two provinces where the army has been fighting al Qaeda and its allies.

The source, speaking from the battlefront, also told Reuters that 20 soldiers were wounded in the bombing.

The number of attacks against the army and security forces in the south has risen sharply since Yemen's U.S.-backed army launched its anti-al Qaeda offensive last week.

Western countries fear further destabilisation in Yemen, which is also facing separatists in the south and unrest in the north, could give more space to al Qaeda in the Arabian Peninsula (AQAP), the local branch of the global Islamist militant movement, to plot attacks on international targets.

Washington has used drones to kill members and leaders of the group for several years. In April, a series of air strikes - presumed to be carried out chiefly by U.S. drones - were launched on central and southern provinces of Yemen, which Yemeni authorities said killed some 65 al Qaeda militants.

Saba quoted a military source as saying that most of the militants killed in Sunday's fighting were Saudis, Afghans, Somalis and Chechens and other nationalities.

The source also said the army destroyed a number of vehicles and weapons belonging to the militants.

President Abd-Rabbu Mansour Hadi said in a speech last week that 70 percent of al Qaeda fighters in Yemen were foreigners, although AQAP later denied this.

ARMY HAS "BROKEN AL QAEDA BACKBONE"

Late on Saturday, suspected militants in Shabwa threw hand grenades at a central bank building, wounding two soldiers, a local official said.

AQAP and its local ally, Ansar al-Sharia, have been waging an insurgency in southern Yemen for more than three years, battling both government forces and local tribal militias.

Hundreds of people have died in bombings, suicide attacks and commando-style raids by al Qaeda against military and government facilities and foreign nationals.

AQAP's main base has been the mountainous al-Mahfad area in Abyan where militants fled in 2012 after the army, with the help of U.S. drones, ousted them from towns they had seized during the chaotic uprising against then-president Ali Abdullah Saleh.

Yemeni defence minister Mohammed Nasser Ahmed was quoted as telling Kuwaiti newspaper Al-Seyassah that the army's offensive in Shabwa and Abyan had "broken the backbone of the al Qaeda terrorist group and destroyed a number of its strongholds, equipment and weapons". (Reporting by Mohammed Mukhashaf; Writing by Yara Bayoumy; Editing by Sami Aboudi and Raissa Kasolowsky)